Covariant star product on semi-conformally flat noncommutative Calabi–Yau manifolds and noncommutative topological index theorem

Abstract

A differential geometric statement of the noncommutative topological index theorem is worked out for covariant star products on noncommutative vector bundles. To start, a noncommutative manifold is considered as a product space [Formula: see text], wherein [Formula: see text] is a closed manifold, and [Formula: see text] is a flat Calabi–Yau [Formula: see text]-fold. Also, a semi-conformally flat metric is considered for [Formula: see text] which leads to a dynamical noncommutative spacetime from the viewpoint of noncommutative gravity. Based on the Kahler form of [Formula: see text] the noncommutative star product is defined covariantly on vector bundles over [Formula: see text]. This covariant star product leads to the celebrated Groenewold–Moyal product for trivial vector bundles and their flat connections, such as [Formula: see text]. Hereby, the noncommutative characteristic classes are defined properly and the noncommutative Chern–Weil theory is established by considering the covariant star product and the superconnection formalism. Finally, the index of the ⋆-noncommutative version of elliptic operators is studied and the noncommutative topological index theorem is stated accordingly.